A variation on the LBGK is the ELBGK [1]}. In this family of
methods, the equilibria are defined as the conditional entropy
maximizers under given values of macroscopic variables (entropic equilibria). The entropies have been constructed in a
lattice dependent fashion in [2]}. A slightly
different notation is used for the lattice Boltzmann algorithm,
\(f_i( \mathbf {x} + \epsilon \mathbf {v}_i,t+\epsilon ) = f_i( \mathbf {x},t) +\alpha \beta (f^{\mathrm {eq}}_i( \mathbf {x},t) - f_i( \mathbf {x},t) ).\) 
